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\title{\textbf{Social Network Analysis of the Reservation System in India
}}
\author{Sudarshan Iyengar}
\institute{IIT Ropar}
\date{\today}
\begin{document}
\maketitle
\begin{abstract}
Many people argue that the reservation system in India, which has existed since the time of Independence, has caused more havoc and degradation of the Indian community than progress. However, these notions have not been based upon any scientific study or research, but have been the result of radical feelings propagated by a few so called "leaders". In this paper, we revisit the cultural divide among the Indian population on a purely network based approach. We study the reservation system in detail, starting from it's past and observing it's effect on the people. Through a survey, we analyze the variation in behavioral characteristics exhibited towards members of the same caste group versus members from another caste group. We use a detailed mathematical model to study various network characteristics. We study the distinct cluster formation that takes place in the Indian community, and find that this is largely contingent on the caste of a person. We define a term benefit, and study the net benefit possessed by each of these clusters. We also study the changes that take place with regard to cumulative benefit of a cluster as well as network formation when a new link is established between the clusters which in its essence, is what the reservation system is doing. Our extensive study calls for reinterpretation of the minds of the Indian people. Although the animosity towards the reservation system could be rooted due to historical influence, hero worship and herd mentality, our results make clear that the system does nothing but aid in the growth of the country and try to bridge the gap between these conflicting social groups.
\end{abstract}


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%\keywords{Social Networks | Friendship Links | Strength of Ties | Caste Based Reservation}

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%\dropcap{I}n this article we study the evolution of ''almost-sharp'' fronts

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The Indian society has always been intrinsically linked to the caste system. This system has been deeply rooted into the blood of the Indians since time immemorial. Caste is defined to be �\emph{a Hindu hereditary class of socially equal persons, united in religion and usually following similar occupations, distinguished from other castes in the hierarchy by its relative degree of purity or pollution}\cite{def-caste}.�It is said that the origin of the caste system is credited to the \emph{Vedas}, which are the very basis of the Hindu religion. According the Rig Veda, the primal man - \emph{Purush} - destroyed himself to create a human society. The different \emph{Varnas}, or castes were created from different parts of his body. The Brahmans were created from his head; the \emph{Kshatrias} from his hands; the \emph{Vaishias} from his thighs and the \emph{Sudras} from his feet. The \emph{Varna} hierarchy is determined by the descending order of the different organs from which the \emph{Varnas} were created. Other religious theory claims that the \emph{Varnas} were created from the body organs of Brahma, who is the creator of the world. This stratification has stayed in society since time itself. Unfortunately, a system that was thought to be designed for organization and efficient labor handling within a society backfired completely the moment people were condemned by their caste. The ugly side of this system surfaced with the birth of untouchability, which is still a dominant phenomenon today. As time progressed, caste became intrinsically linked with one's wealth, social status and even entry into public places. The society became largely dominated by the upper castes, and the other \emph{varnas} were forced into submission while one section of the society became unbearably wealthy and powerful.

To offset this heinous societal structure, the system of untouchability was abolished, and the Reservation system was introduced to provide the largely ignorant socially backward community of the country with basic opportunities such as education and jobs. However it was soon met with a lot of backlash from the socially forward community, who felt they were being unjustly treated by allowing for special benefits for the downtrodden.
In this paper, we study the existing system in detail. We study the two divided communities, the \emph{socially forward and uplifted} (SF) and the \emph{socially backward and downtrodden} (SB). We establish why and how the reservation system maintains a very good balance between the two, and is only taking the country further every day. Primarily, we justify the need for this system to continue to exist in present day India, by means of a simple survey, and by gathering certain relevant data from a nation-wide census. The mathematical model aids us in finding many parameters, which quantify upliftment and betterment of the country. In addition we consider the number of links between the two communities as a measure of stability of the large scale social structure. We quantify the \emph{forward breeze effect}, which is, to put simply, the change in mind-set of the SF, on being in contact with the SB. We also quantify the \emph{reverse breeze effect}, which is the increased motivation felt by the SB to achieve upliftment, by being influenced by those close to them and around them. We study the cumulative benefit of each cluster in discrete time steps and observe how it changes when this system is in place.

Our motivation comes from the existence of a tangible strength associated with every weak tie, as proposed my Mark S.Granovetter in his famously cited theory of the Strength of Weak Ties \cite{sowt}. This is a very prominent network phenomenon has been ignored in past studies of this system, which we have chosen as our key element of study. Past studies in network analytics by renowned social network theory expert M. O. Jackson has shown  that network formation and subsequent interaction between the nodes is highly influenced by homophily \cite{homo}\cite{new-3} . In India, associations amongst the people is seen to be largely caste based, hence for the purposes of our study, we choose to term the network formation pattern among the Indian population as a \emph{caste-based homophilic network}. We have studied this phenomenon using a simple survey to show friendship ties, and found that majority of the links existed among the same caste groups. Such a network structure and the resulting non-cooperative behavior is the basis of discrimination, which is detrimental to harmonious growth of the country.
\begin{table}
\caption{Results of the Survey conducted among students of a local high school to depict friendship ties : Each subject was asked to name four close friends and their social groups}\label{*}
\begin{tabular}{@{\vrule height 10.5pt depth4pt  width0pt}ccccc}
\hline
Criteria & Percentage of Survey Population \\
\hline
All four ties from within the same social group & 53\\
Three ties from the same social group & 19\\
Two ties from the same social group & 20\\
One tie from the same social group & 6\\
All ties from different social groups & 2\\
\hline
\end{tabular}
\end{table}
In this paper we consider a modified but natural model where it is common knowledge that the state of the world changes deterministically over time, as new network connections are added through time steps. As our main contribution, we introduce in this paper the prominent role played by strength of weak ties in alleviating the divide between the caste groups in the Indian scenario. By our network model, we find it sufficient to insert a minimal number of links between the two clusters, in order to foster harmonic relations between the two conflicting groups. We note the importance of propagation of benefits through weak ties between the clusters. A member from SB befriending a member from SF results in a whirlwind propagation of benefits through the single link between the clusters, and this propagation is studied in detail through our model. We also note that benefit does not propagate fully from one node to another, but as a decay centrality function through friendship ties, which has been previously studied in detail by Mathew O. Jackson in his book, Social and Economic Networks \cite{decay}. Here we study the variation of this network parameter particular to a cluster, \emph{propagation factor}, which is the proportion of benefit propagated from one node to another. As a long term aim of the Reservation System, we see a reasonable balance of benefits distributed evenly among members from the forward as well as the backward communities. The current statistics show a clear tip in the balance favoring the socially forward community, with a majority of the country's shared resources such as education, wealth and land-holdings being in the possession of or being accessible to only one section of the society. This undesirable disparity can be seen as the result of many recent studies in this matter, including the works of A. K. Shiva Kumar and Preet Rustagi \cite{disp-1}, Mona Sedwal and Sangeeta Kamat \cite{disp-2} and R R Biradar \cite{isec}. Hence, on studying this system in detail, analysing the changes in individual benefits that result due to addition of edges between the clusters through the reservation system and watching the strength of weak ties in play in propagating benefits even to those who do not directly benefit from the Reservation System, we note a number of striking changes that arise in individual welfare. The effect of the absence of the system cannot be studied in the present day scenario due to the extant and obvious reasons, however a similar study has been performed by Vani Borooah, Dubey K, Amaresh and Sriya Iyer \cite{sriyaiyer}. The study takes into account a social group within the country which was at the same social, educational and economic status as the Scheduled Castes and Scheduled Tribes in the pre-independence era. However, the condition of this group was observed to be the same if not worse during their study conducted in 2009, while the SC's and ST's were found to be at a much more elevated state, proving the efficiency of the Reservation System. Many such similar studies show clearly the tangible upliftment experienced by the Scheduled Castes and Scheduled Tribes, and such comparative studies along with extant evidence only bolster the claim that this system is indeed a beacon of hope for the social disparity in India.

However, in spite of being in a far better position than they were in the pre-independence era, the socially backward community is however still disadvantaged with respect to institutions such as education and economy. Thomas E. Weisskopf notes in his paper \cite{new-7} that there still exists unequal representations in educational institutions in terms of social class, with the forward community being dominant and occupying a majority of the seats in spite of being far fewer in terms of population\cite{samp-1,samp-2}. The proportion of graduates from backward communities were also found to be extremely low \cite{new-7}. On studying the performance in nation-wide examinations, it was found that a very small proportion of backward classes performed on par with the forward classes \cite{samp-4}. On assessing the academic performance of the beneficiaries of reservation it was found that an unfavorable minority of them manage to graduate degree programs and qualify as a graduate\cite{samp-10,samp-11}. A major hurdle that needs to be necessarily overcome is the incomplete filling up of reserved seats\cite{samp-6,samp-7,samp-8}, which shows the overwhelming need to change the mind set of the socially backward community from opting for low paying daily wage labor to a secure educational degree. 
Most of these studies involved participants who were first time beneficiaries of reservation. Studies on reservation for second level beneficiaries\cite{samp-9} show that their performance is visibly better than their social counterparts who are first level beneficiaries, leading us to conclude that reservation must indeed exist for at least two generations, if not more in order to show a marked improvement in status.


\section*{Model and Preliminaries}
\subsubsection*{Network Structure}
In this subsection, we aim to define our network model and analyze the working of the system in detail. The structure of the network is dynamic and is modeled by a finite undirected graph defined as $G = (V, E)$ where $V$ is a finite set of nodes and $E$ is the set of edges between two nodes. Each node $v\in V$ represents an individual in a community, while each undirected edge $e = (u, v)\in E$ represents a tie between two individuals.\\
\\
We initially define $G$ to consist of two distinct, disconnected clusters of nodes representing the two communities under consideration: the socially forward and uplifted community and the socially backward and downtrodden community. We note that due to a history of oppression and disparity between these social groups, these two clusters remain disconnected at the time of Independence, which is not a bad conclusion to make. Each cluster is generated as an individual \textit{Erd$\ddot{o}$s-Renyi} graph $G_{ER} = (n, p')$ \cite{erdos}, where edges between any two of the $n$ nodes are added with probability $p'$ independent from every other edge.
In an Erd$\ddot{o}$s-Renyi graph the total number of edges is given by

\begin{equation}
|E| =  \frac{n (n-1)p}{2'}
\end{equation}

Since the two clusters in $G$ are disconnected at first, no $e = (u, v) \in E$ exists, such that $u$ and $v$ belong to different clusters, as shown in Fig.~\ref{cluster1}.\\
\\
We define the socially uplifted community cluster as $FC = \{i : i \in V$\} and the socially backward community cluster as  $BC =  \{j : j \in V$ and $j \not\in FC\}$ such that $FC \cap BC = \phi$, the number of nodes $n_G = |V|$, the total number of inter-cluster edges $n_{inter} = |\{e = (u, v): u \in FC, v \in BC\}| $ and total number of within-cluster edges $n_{within} = |\{e = (u, v): u \in FC, v \in FC\} \cup \{e = (u, v): u \in BC, v \in BC\}| $.\\
\\
We now define a key term known as the \emph{propagation factor} $p \in [0, 1]$ which is the percentage of benefit intrinsic to one node that propagates to its neighbors as a result of friendship ties between them. The propagation factor corresponds to how likely an agent is to pass on its access to shared resources to its neighbors. The special case $p = 1$ represents the maximum cooperation possible in a network, where the agent's neighbors have access to everything that the agent itself has access to. Likewise, the other extreme case $p = 0$ represents the maximum degree of non-cooperation in a network, wherein the agent's neighbors do not benefit at all by maintaining a friendship link with the agent.\\
\\
In this paper, we use the terms \emph{benefit} and \emph{influence} interchangeably, as they depict the same phenomenon in our study. In order to show the actual working of the reservation system, we define an additional variable %$k = |{n \in F}|$ to be the number of nodes from the FC cluster connected to a node in BC cluster,
depicting a proposed and realistic situation wherein a member of the backward community who benefits from reservation has a chance to interact with some members of the forward community. We model this interaction and tie-formation as $k$ which is the number of nodes from $FC$ that form ties with a single node in $BC$.
We also define the benefit possessed by each node $i \in FC$ as $B_i = \Delta$, the benefit possessed by each node $j \in BC$ as $B_j = 0$ initially and the distance between two nodes $u, v$ as $l_{uv}$. For any two nodes $u$ and $v$, the distance between the two nodes is the number of edges on the shortest path connecting the two of them. Finally, we define the cumulative influence of the $BC$ cluster as
\begin{center}
\begin{equation}
B = \sum_{j \in BC} B_j
\end{equation}
\end{center}


\subsubsection*{Network Formation and Influence Propagation}
Initially the two clusters $FC$ and $BC$ are entirely disconnected. This means that $l_{uv}$ for $u \in FC$ and $v \in BC$ is initially infinite.

At each iteration, $k$ edges are added between some randomly chosen $u \in BC$ and $k$ randomly chosen nodes, $\{v_1, v_2, ..., v_k\} \subset FC$. Hence, the total number of inter-cluster edges now becomes
\begin{equation}
n_{inter}=\frac{1}{2}(n_G*p_e)
\end{equation}
where $p_e$ is the percentage of nodes that are connected from $BC$.\\
\\
The cumulative benefit possessed by the socially backward community, $B$, depends on the propagation factor $p$ and the number of edges $n_{inter}$.
Our motivation towards this line of thinking is that complete knowledge about influence factors can be assumed in a uniform social network \cite{conc1} and also that related actions of members of a social network can help to estimate the influence factors \cite{conc2}. We assume here that the density of both clusters are same, i.e. both the clusters have equal probability $p'$ in forming within-cluster edges.

From \cite{decay}, the benefit of an agent propagates through the network in the order $p$ to its immediate neighbor, $p^2$ to the next level of neighbors and so on. This conclusion is in line with the concept of \emph{decay centrality}, which plays a major role in determining many parameters linked to network analytics. Therefore, when an edge $e = (u, v)$ first exists such that $u \in FC, v \in BC$, the benefit gained by $v$ due to $u$ is given by
\begin{equation}
B_v = B_u \times p = \Delta \times p
\end{equation}

The neighbors of $v$ also gain some amount of benefit through $v$'s tie with $u$. For some $w \in BC$ such that $e = (w, v)$, its benefit is now given by
\begin{equation}
B_w = B_v \times p = \Delta \times p^2 = \Delta \times p^{l_{wu}}
\end{equation}

Similarly, all neighbors of $w$ will gain some influence, and so on.
For the purposes of our study, we assume that influence beyond two friendship levels of the socially forward community does not propagate to the socially backward community, due to the propagation factor becoming negligible. However, we assume that any small gain in influence will propagate through the entire socially backward community network.

This key point of observation has been neglected in many previous studies of disparate human social networks, wherein adding a single link between disjoint clusters helps to uplift not only the agent who is directly a part of the link, but also the rest of the agent's community through the propagation of benefits by the \emph{Strength of Weak Ties}.
\begin{list}{•}{•}
\item[$\bullet$ At $n_{inter}=0$] The initial state is as shown in Fig.~\ref{cluster1}. At this stage, $B_u = 0$ $\forall u \in BC$. Since there is a non-existent path to reach any node in $BC$ from any node in $FC$, there is no propagation of influence from $FC$ cluster to $BC$ cluster. Hence the net influence of $BC$ cluster $B = \sum_{u \in BC} B_u = 0$
\begin{figure}[!h]
\begin{center}
\begin{subfigure}{0.3\textwidth}
\centering
\includegraphics[width=0.8\linewidth]{cluster1.png}
\caption{Zero inter-cluster edges}
\end{subfigure}%
\begin{subfigure}{0.3\textwidth}
\centering
\includegraphics[width=0.8\linewidth]{cluster2.png}
\caption{One inter-cluster edge}
\end{subfigure}%
\begin{subfigure}{0.3\textwidth}
\centering
\includegraphics[width=0.8\linewidth]{cluster3.png}
\caption{Multiple inter-cluster edges}
\end{subfigure}
\caption{Disparate Disconnected Clusters}
\label{cluster1}
\end{center}
\end{figure}
\item[$\bullet$ At $n_{inter}>0$] When $k$ edges are added between some node $u\in BC$ and $k$ randomly chosen nodes, $\{v_1, v_2, ..., v_k\} \subset FC$, the value of $B_u$ increases multifold. Eventually, as explained above, $B_u$ also propagates to $u$'s neighbors. Fewer the number of inter-cluster edges, higher is the average number of hops to reach some $v' \in FC$ from some $u' \in BC$. The addition of edges between these clusters reduces the average $l_{u'v'}$ by a great degree. Hence the cumulative benefit possessed by $BC$, $B$, increases as average $l_{u'v'}$ decreases. $B$ also depends on the propagation factor, $p$, which indirectly refers to the strength of the network. The propagation factor determines the proportion of $\Delta$ that is propagated to within the backward community. Many recent studies on the Indian social structure in particular show that network strength is highest in communities with weakest outside options \cite{diamond}. This is especially relevant to this study as the focus lies on the socially backward communities such as Scheduled Castes and Scheduled Tribes, who have been stuck in occupational and poverty traps for centuries together \cite{isec}, and hence have developed a highly intraconnected and cooperative social network. In order to increase $B$, average shortest path length between $FC$ and $BC$ clusters must show a significant decrease and the propagation factor must increase. The first ensures more direct access to resources, and the second ensures greater influence propagation where access is indirect.
\end{list}
We perform detailed experiment on the above network variables, by studying the variation $p$ for a range of values between $0$ and $1$. Initially we vary $p$ between 0.1 and 0.8 with discrete steps of 0.1, and then we subsequently vary $p$ between 0.001 and 0.008 in discrete steps of 0.001, in order to obtain more clarity on the underlying process.
For every new value of $p$ we also study the change due to increasing number of inter-cluster edges.


\subsection*{Observations}
\begin{figure}[!h]
\includegraphics[width=1\textwidth]{2DPlot1.png}
\caption{Increase in Cumulative Influence as a function of number of edges and propagation factor for p = 0.1 and p = 0.2}
\label{plot1}
\end{figure}

In the first case where $p$ was varied between 0.1 and 0.8, we isolated the first two sets of observations and plotted two line graphs with the variables being number of edges and cumulative influence of the socially backward community for two different values of $p$, 0.1(in red) and 0.2 (in blue) as shown in Fig.~\ref{plot1}.

\begin{itemize}
  \item Both the plots were found to be more or less linear, with the plot of $p = 0.1$ being significantly more linear than the plot for $p = 0.2$

  \item There is nearly a ten-fold initial difference in the cumulative benefit, even when $n_{inter} = 1$.
  Whereas the initial cumulative influence is 35.19 for $p = 0.1$, it is 305.09 for $p = 0.2$
This leads us to the conclusion that with just one-tenth increase in propagation, the socially backward community is benefitted 10 times as much, even when there is only one tie between the communities.

  \item Our studies also found that there is a significant difference in the slopes of the two line plots.
By the process of linear fitting, the obtained line equations for the two line plots were as follows:
\begin {itemize}
\item For $p = 0.1: y = 1.8x + 43$
\item For $p = 0.2: y = 8.2x + 3.7e+002$
\end{itemize}
This shows that with one-tenth more propagation, the rate with which benefits accumulate as we increase percentage of reservation, increases nearly four-folds.
The increase in percentage of reservation does not seem to affect the cumulative influence as much as the increase in propagation of gained influence does.

  \item While the plot for $p = 0.1$ is more linear, the plot for $p = 0.2$ shows some patterns of initial acceleration before the rate of increase of cumulative benefit peters out to become more or less linear. Apart from the slope difference observed above, it seems that coupled with a high propagation factor, rate of increase of influence shows a steep climb when the first 6-8 edges are added.
\end{itemize}

\begin{figure}[!ht]
\begin{center}
\includegraphics[width=1\textwidth]{2DPlot2.png}
\caption{Increase in Cumulative Influence as a function of number of edges and propagation factor for p = 0.001 to p = 0.008}
\label{plot2}
\end{center}
\end{figure}

\begin{figure}[!ht]
\begin{center}
\includegraphics[width=1\textwidth]{2DPlot3.png}
\caption{Increase in Cumulative Influence as a function of number of edges and propagation factor for p = 0.001 and p = 0.002}
\label{plot3}
\end{center}
\end{figure}

In the second case where $p$ was varied between 0.001 and 0.008, all 8 sets of observations were initially plotted as seen in Fig.~\ref{plot2}. Subsequently, the observations for the first two values of $p$ were isolated, and the two line graphs with the variables being number of edges versus cumulative influence of backward community were plotted for two different $p$ values of 0.001(in red) and 0.002 (in blue) as shown in Fig.~\ref{plot3}.
\begin{itemize}
\item Both the plots were found to be roughly straight line plots.

\item From these, it was found that there is nearly a four-fold initial difference in the cumulative influence, even when $n_{inter} = 1$.
Whereas the initial cumulative influence is $0.0015$ for $p = 0.001$, it is $0.0058$ for $p = 0.002$
This means that with just one-thousandth more propagation, the backward community is benefited 4 times as much, even when there is only one tie between the communities.

\item There is a significant difference in the slopes of the two line plots.
After performing linear fitting, the line equations that we obtained for the two line plots were as follows:
\begin {itemize}
\item For $p = 0.001: y = 0.0012x + 0.0021$
\item For $p = 0.002: y = 0.0042x + 0.0085$
\end{itemize}
This implies that with one-thousandth more propagation, the rate with which benefits accumulate as we increase reservation, increases nearly four-folds.
This again leads us to the conclusion that the increase in percentage of reservation does not affect the cumulative influence as much as the increase in propagation of gained influence does.

\end{itemize}
\section*{Related Works}
Social disparity is a common feature in many countries, and was prominently noticed in the American continents in the form of discrimination against African Americans or the Blacks. A lot of study has been put into perspective in order to gain a better understanding of this phenomenon. A comprehensive study by M. O. Jackson on Employment and Wages \cite{new-1} showed that social networks played a major role in determining employment and employability. Groups which remained unemployed found it more difficult to get employment, than those which were already into some form of labor. Studies on black-white links also showed that a great social distance was present between members of the two communities, and this resulted in a wage gap unfavorably inclined towards the blacks who were seen as the disadvantaged community \cite{new-2}. Studies have also been conducted on the role of social networks in determining an individual's nature of job and wages, and have found that individuals with the same capability end up with very differently ranged incomes due to the ties or friendship links they may have \cite{new-4}, which is a phenomena that we revisit in this paper. Apart from affirmative action in the American continents, discrimination has also been extensively been studied in Europe \cite{europe} using the multi-national European Social Survey, which quantitatively measured attitudes and feeling using questionnaires, and resulted in similar thesis.
 
 Homophily, which is an important network parameter being considered in this paper, has been dealt with in detail in many past studies \cite{homo,new-3,new-5, new-6} and it's effects on human social networks have been studied by numerous network experts. These studies have convincingly concluded that homophily plays a determining role in the formation, maintenance and decline of human social networks. In addition, homophily is also seen to heavily influence the spread of information and cooperation through a social network, which has again been analyzed in detail by Nicholas A.Christakis in \cite{ch-1, ch-2}. Also,the study of cooperation in dynamic networks \cite{ch-3}, the cascading effects of cooperation in a network \cite{rw-1}, and the various factors determining variation in cooperation across populations \cite{shakti} are important in analysing real world networks such as in the Indian scenario, and such studies give a better perspective into networks as a subject in analyzing and determining human behavior in social networks.
 
Small world networks play a crucial role in this particular study as we study two disparate real-world small world networks, and past researches conducted on this topic by Duncan J. Watts and Steven H. Strogatz \cite{sw-1} and Mark D. Humphries and Kevin Gurney \cite{sw-2} analyze many parameters related to small worlds, and help in identifying and classifying them effectively, aiding this study in a significant way.



\section*{Conclusion}
In this ground-breaking and much needed study of a highly controversial and complex social network, we have studied a diffusion process depicting propagation of benefits to a community that has been oppressed for time immemorial. We found that the Reservation System works in mystifying ways, by uplifting a socially downtrodden community not just by allowing for direct interactions between two non-interacting elements of the Indian society, but by also allowing for this interaction to lead to tangible benefits to every other member of the downtrodden community by making effective use of the Strength of Weak Ties, a phenomenon which has not been looked at in this angle ever before. Towards understanding the necessity and functioning of the Reservation System through a network based analysis, we conducted a survey which established that the disparity has not ended and thus a bridging system to interlink the clusters is still needed today.  In this study we focussed on the influence mechanism that propagates through friendship ties and also directly depends on the \emph{Propagation Factor}, an indicator of network strength. We studied the individual benefit possessed by each agent in a cluster, and found that in order to bring the net individual benefit of one cluster on par with another, a clever combination of links between the clusters and cooperation within the cluster in the form of propagation of benefits need to co-exist. We conclude by saying that, Reservation System is one such system which combines these two determining parameters and ensures slow but guaranteed upliftment of every member of the downtrodden community.

As a part of future work, we plan on investigating a larger variety of social network topologies wherein such a disparity exists, and apply a similar system in order to test its efficiency. This will allow us to further analyze the varying effects of propagation factor, percentage of reservation as well as number of edges added between the clusters, and find the exact link between the various parameters. Additionally, we plan on studying the social structure within the Indian subcontinent in greater detail, and arrive at a specific percentage of reservation, which we term as the \emph{Ideal/Optimum Number, }which if applied, will cause great amount of upliftment within shortest time, keeping in mind a new parameter known as \emph{Social Distance}.
%



%%%%%%%%%% BIBLIOGRAPHY %%%%%%%%%%
\begin{thebibliography}{99}
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\end{thebibliography}

%\begin{description}
%\item{erdos-renyi}
%Erdos, Paul, and Alfred Renyi. "On the strength of connectedness of a random graph." Acta Mathematica Hungarica 12.1 %(1961): 261-267.
%\end{description}

\end{document}
